You're absolutely right.
Christianity didn't care about wealth, gender, nation, or social class, factors which dominated the societies of the time. It had many attractions for the dispossessed, which, of course, comprised the vast majority of the population. This was an enormous selling point, and a real philosophical advance over every other belief system of its time. The Christian God did not care who you were, just what you thought. And everyone can control his own thoughts. It became an unstoppable meme.